Approximately Piecewise E(3) Equivariant Point Networks (2402.08529v1)

Abstract: Integrating a notion of symmetry into point cloud neural networks is a provably effective way to improve their generalization capability. Of particular interest are $E(3)$ equivariant point cloud networks where Euclidean transformations applied to the inputs are preserved in the outputs. Recent efforts aim to extend networks that are $E(3)$ equivariant, to accommodate inputs made of multiple parts, each of which exhibits local $E(3)$ symmetry. In practical settings, however, the partitioning into individually transforming regions is unknown a priori. Errors in the partition prediction would unavoidably map to errors in respecting the true input symmetry. Past works have proposed different ways to predict the partition, which may exhibit uncontrolled errors in their ability to maintain equivariance to the actual partition. To this end, we introduce APEN: a general framework for constructing approximate piecewise-$E(3)$ equivariant point networks. Our primary insight is that functions that are equivariant with respect to a finer partition will also maintain equivariance in relation to the true partition. Leveraging this observation, we propose a design where the equivariance approximation error at each layers can be bounded solely in terms of (i) uncertainty quantification of the partition prediction, and (ii) bounds on the probability of failing to suggest a proper subpartition of the ground truth one. We demonstrate the effectiveness of APEN using two data types exemplifying part-based symmetry: (i) real-world scans of room scenes containing multiple furniture-type objects; and, (ii) human motions, characterized by articulated parts exhibiting rigid movement. Our empirical results demonstrate the advantage of integrating piecewise $E(3)$ symmetry into network design, showing a distinct improvement in generalization compared to prior works for both classification and segmentation tasks.

• The paper introduces APEN, a neural network design that achieves approximately piecewise E(3) equivariance for improved point cloud analysis.
• The methodology uses a multi-layer partitioning strategy that progressively approximates global E(3) symmetry without relying on predefined partitions.
• Empirical results on room scans and human motion data demonstrate significant improvements in classification and segmentation accuracy.

An Overview of Approximately Piecewise E(3) Equivariant Point Networks

This paper introduces a framework for designing neural networks that are approximately piecewise E(3) equivariant, specifically aimed at point cloud data. The critical importance of incorporating E(3) symmetry, which refers to the group of Euclidean transformations, in point cloud neural networks is underscored for its ability to enhance generalization. Previously, research has focused on networks that are equivariant under global E(3) transformations. However, real-world scenarios often involve complex systems composed of multiple independently transforming parts. This paper addresses the challenge of maintaining E(3) equivariance in such systems by introducing the APEN framework.

Technical Approach

APEN employs a multi-layer compositional design that starts with a fine partition of the point cloud, progressively transitioning towards coarser partitions. Each layer in this architecture maintains piecewise equivariance concerning a finer partition, leading to successive layers that approximate the true partition more accurately. The network achieves this without pre-defined partitions, which are usually unknown a priori in practical applications.

The core of the methodology lies in the knowledge that having a finer partition that includes the true partition ensures equivariance. This insight is foundational to APEN's partition prediction model, which incrementally builds a proper subpartition. The model measures the equivariance approximation error based on two key factors: the uncertainty in the partition prediction and the likelihood of failing to suggest a correct subpartition.

Empirical Results

The paper presents empirical evidence showcasing the efficacy of APEN. Two datasets are used: (i) real-world room scans with multiple object types, and (ii) human motion data where articulated parts exhibit movement. The results demonstrate that integrating piecewise E(3) symmetry significantly enhances generalization accuracy for both classification and segmentation tasks compared to existing methods.

Implications and Future Work

APEN represents an advancement in designing neural networks capable of handling the intricacies of piecewise E(3) symmetrical point cloud data. The proposed framework not only improves task-specific performance but also provides a structured approach to understanding equivariance approximation errors, effectively bridging the gap between theoretical properties of symmetry and practical implementation in neural networks.

Looking ahead, there is potential to expand this framework to other tasks within AI, such as generative modeling and 3D reconstruction, where the benefits of incorporating symmetries could further improve system performance. Furthermore, the theoretical exploration of Partially E(3) Equivariant networks concerning error bounds remains a fertile area for future research.

Ultimately, this paper contributes a significant methodological development in the landscape of point cloud neural networks, offering insightful pathways for both theoretical and practical enhancement of 3D recognition and processing systems.